Geometrical Methods in Goursat Categories
Abstract
The main aim of the paper is to show that the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3, hold in any regular Mal'tsev categories. We prove that Mal'tsev categories may be characterized through variations of the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3, that is classically expressed in terms of four congruences R, S1, S2 and T, and characterizes congruence modular varieties. The proof of this result in a varietal context may be obtained exclusively through the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3. This was shown by H.P. Gumm in Geometric Methods in Congruence Modular Algebras. We prove that for any 2n+1-permutable category $\mathcal{E}$, the category Equiv$(\mathcal{E})$ of equivalence relations in $\mathcal{E}$ is also a 2n+1-permutable category.
References
B. R. Amougou Mbarga, Triangular scheme revisited in the light of n-permutable categories, Earthline Journal of Mathematical Sciences 6(1) (2021), 105-116. https://doi.org/10.34198/ejms.6121.105116
B. R. Amougou Mbarga, Anticommutativity and n-schemes, Earthline Journal of Mathematical Sciences 6(1) (2021), 175-186. https://doi.org/10.34198/ejms.6121.175186
B.R. Amougou Mbarga, Some remarks on Goursat lemma, Algebraic Structures and their Applications, in Press.
D. Bourn, Internal equivalence relations, modular formula and Goursat condition in non-regular context, Cahiers du L.M.P.A., Published online: http://www-lmpa.univ-littoral.fr/publications/articles/lmpa526.pdf
M. Barr, Exact categories, in: Exact Categories and Categories of Sheaves, 1-120, Lecture Notes in Math., vol. 236, Springer, Berlin, 1971. https://doi.org/10.1007/BFb0058580
F. Borceux and D. Bourn, Metatheorems, in: Mal'cev, Protomodular, Homological and Semi-Abelian Categories, Mathematics and its Applications, vol. 566, Kluwer Academic Publishers, Dordrecht, 2004. https://doi.org/10.1007/978-1-4020-1962-3_1
S. Burris and H.P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics, 78, Springer-Verlag, New York-Berlin, 1981.
A. Carboni, J. Lambek and M. C. Pedicchio, Diagram chasing in Mal'cev categories, Journal of Pure and Applied Algebra 69 (1991), 271-284. https://doi.org/10.1016/0022-4049(91)90022-T
A. Carboni, G. M. Kelly and M. C. Pedicchio, Some remarks on Maltsev and Goursat categories, Appl. Categor. Struct. 1 (1993), 385-421. https://doi.org/10.1007/BF00872942
I. Chajda and E.K. Horváth, A scheme for congruence semidistributivity, Discuss. Math. Gen. Algebra Appl. 23 (2003), 13-18. https://doi.org/10.7151/dmgaa.1060
I. Chajda, E.K. Horváth and G. Czédli, Trapezoid lemma and congruence distributivity, Math. Slovaca 53 (2003), 247-253.
J. Lambek, Goursats Theorem and the Zassenhaus Lemma, Canad. J. Math. 10 (1958), 45-56. https://doi.org/10.4153/CJM-1958-005-6
M. Gran, Notes on regular, exact and additive categories, Summer School on Category Theory and Algebraic Topology, Ecole Polytechnique Fédérale de Lausanne, 11-13 September 2014.
M. Gran and D. Rodelo, Beck-Chevalley condition and Goursat categories, 2013. arxiv:1512.04066v1
M. Gran, D. Rodelo and I. T. Nguefeu, Variations of the shifting lemma and Goursat categories, Algebra Univers. 80 (2019), Paper No. 2. https://doi.org/10.1007/s00012-018-0575-z
M. Gran, D. Rodelo and I. Tchoffo Nguefeu, Facets of congruence distributivity in Goursat categories, 2020. https://arxiv.org/abs/1909.10211v2
M. Gran, D. Rodelo and Idriss Tchoffo Nguefeu, Some remarks on connectors and groupoids in Goursat categories, Logical Methods in Computer Science 13(3) (2017), 1-12. https://doi.org/10.23638/LMCS-13(3:14)2017
B. Jonnson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110-121. https://doi.org/10.7146/math.scand.a-10850
J. Hagemann and A. Mitschke, On n-permutable congruences, Algebra Univers. 3 (1973), Article number: 8. https://doi.org/10.1007/BF02945100
M. Hoefnagel, Anticommutativity and the triangular lemma, 2020. arXiv:2008.00486v2
M. Hoefnagel, Majority categories, Theory Appl. Categ. 34 (2019), 249-268.
M. Hoefnagel, Characterizations of majority categories, Appl. Categor. Struct. 28 (2020), 113-134. https://doi.org/10.1007/s10485-019-09571-z
M. Hoefnagel, A categorical approach to lattice-like structures, Ph.D. thesis, 2018.
P.-A. Jacqmin and D. Rodelo, Stability properties characterising n-permutable categories, Theory and Applications of Categories 32(45) (2017), 1563-1587.
K.A. Kearnes and E.W. Kiss, The triangular principle is equivalent to the triangular scheme, Algebra Univers. 54 (2005), 373-383. https://doi.org/10.1007/s00012-005-1954-9
Mathilde Olivette Ngaha Ngaha, Isomorphism theorems and descent in star-regular categories, Prom.: Gran, Marino. http://hdl.handle.net/2078.1/145867
H. Peter Gumm, Geometrical methods in congruence modular algebras, Mem. Amer. Math. Soc. 45 (1983). http://dx.doi.org/10.1090/memo/0286
H. Peter Gumm, The little Desarguesian theorem for algebras in modular varieties, Proc. Amer. Math. Soc. 80 (1980), 393-397. https://doi.org/10.1090/S0002-9939-1980-0580991-6
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg){kind=logo}
